Autocovariance Estimation in Regression with a Discontinuous Signal and m‐Dependent Errors: A Difference‐Based Approach. (16th November 2016)

• Record Type: Journal Article
• Authors: Tecuapetla‐Gómez, Inder; Munk, Axel
• Is Part Of: Scandinavian journal of statistics. Volume 44:Number 2(2017:Jun.)
• Page Start: 346
• Page End: 368
• Publication Date: 2016-11-16
• Subjects: autocovariance estimation -- change‐point -- convex projection -- covariance matrix estimation -- difference‐based methods -- discontinuous signal -- m‐dependent processes -- mean squared error -- non‐parametric regression; Statistics -- Periodicals; 310
• DOI: 10.1111/sjos.12256
• Languages: English
• ISSNs: 0303-6898

Abstract

Abstract: We discuss a class of difference‐based estimators for the autocovariance in nonparametric regression when the signal is discontinuous and the errors form a stationary m ‐dependent process. These estimators circumvent the particularly challenging task of pre‐estimating such an unknown regression function. We provide finite‐sample expressions of their mean squared errors for piecewise constant signals and Gaussian errors. Based on this, we derive biased‐optimized estimates that do not depend on the unknown autocovariance structure. Notably, for positively correlated errors, that part of the variance of our estimators that depend on the signal is minimal as well. Further, we provide sufficient conditions for n ‐consistency; this result is extended to piecewise Hölder regression with non‐Gaussian errors. We combine our biased‐optimized autocovariance estimates with a projection‐based approach and derive covariance matrix estimates, a method that is of independent interest. AnR package, several simulations and an application to biophysical measurements complement this paper.